SVI problem

Basic formulation

The SVI test is a multiparametric problem. Usually, a polytropic gas with a specific heat ratio of γ = 1.4 is considered as an object of modeling, the coordinate system is associated with the shock wave (initially, at time t = 0, the shock wave front is stationary and located in the section x = x_S = const), and the two-dimensional vortex is assumed to be isentropic.

The basic parameters of the SVI problem: M_S = u_x / c is the shock-wave Mach number, which characterizes the intensity of the shock wave (u_x is the flow velocity ahead of the shock, c is the sound speed), V = V(r) is the velocity profile of the cylindrical vortex (r is the distance from the vortex center), M_V = V_m / c is the vortex Mach number, which characterizes the intensity of the vortex (V_m is the maximum velocity on the profile). Other parameters of the problem: r₀ is the effective radius of the vortex (determined as the point on the velocity profile, where V(r₀) = V_m), (x_V, y_V) are the coordinates of the vortex center at t = 0.

The computational domain for the SVI problem is typically a rectangle of a unity height in the xy coordinate system. A flow symmetry condition is set at the upper and lower boundaries of the domain, a supersonic inflow is set at the left boundary, and a subsonic outflow at a given pressure (nominal pressure behind the shock wave) is set at the right boundary. The flow is simulated until the time t = t₁, which ensures the complete passage of the vortex through the shock wave and its subsequent significant advancement in the longitudinal direction.

The SVI problem is most often simulated using the Euler equations (inviscid approach) [1–3, 5], but sometimes the Navier–Stokes equations (viscous approach) are used [4, 6, 7]. In the latter case, it is necessary to specify the Reynolds number Re (hereinafter, the Reynolds number is determined by the flow parameters ahead of the shock wave and the effective vortex diameter equal to 2r₀) and the Prandtl number Pr.

From the computational point of view, the most interesting fomulation of the SVI problem corresponds to to the case when an intense vortex passes through a strong shock wave. In this case, the flow pattern has a complex structure with numerous shocks and contact surfaces. However, if this problem is solved within the inviscid approach on a sequence of refining grids, the contact surfaces gradually become unstable, leading to a lack of grid convergence. If the problem is solved within the viscous approach, it is possible to use such value of Reynolds number that preserves the complex structure of the flow and provides the stability of the numerical solution on refining grids.

For numerical method testing, the following formulation of the SVI problem is proposed: γ = 1.4, M_S = 3, M_V = 0.8, Re = 10⁴, Pr = 0.75, x_S = 0, (x_V, y_V) = (–0.5, 0.5), r₀ = 0.075, the vortex velocity profile is defined by the formula

V(r) = V_m ⋅ (r / r₀) ⋅ exp{[1 – (r / r₀)²] / 2},  where r = [(x–x_V)² + (y–y_V)²]^0.5.

The computional domain for basic fomulation of the problem is the rectangle [–1, 1] × [0, 1].

Let us consider in detail the procedure for setting the initial flow fields. For flow parameters, we will use the commonly accepted notation: u_x and u_y are the components of the velocity vector, ρ is the density, and p is the pressure. The subscripts 1 and 2 denote the flow parameters ahead of and behind the shock-wave front, respectively (here we only describe the uniform flow ahead of the shock wave, without accounting for the "vortex" disturbance). By setting ρ₁ = p₁ = 1, we have 

u_x1 = M_S (γp₁ / ρ₁)^0.5 = 3⋅(1.4)^0.5,   u_y1 = 0,

u_x2 = u_x1 ⋅ [(γ–1)M_S² + 2] / [(γ+1)M_S²] = 7⋅(1.4)^0.5/9,   u_y2 = 0,

ρ₂ = ρ₁ ⋅ [(γ+1)M_S²] / [(γ–1)M_S² + 2] = 27/7,   p₂ = p₁ ⋅ [2γM_S² – (γ–1)] / (γ+1) = 31/3.

The initial flow fields are then adjusted by adding a cylindrical vortex to the background flow ahead of the shock wave; the vortex is assumed to rotate clockwise. The adjustment procedure consists of the following steps. First, the maximum velocity on the vortex profile is calculated: V_m = M_V (γp₁ / ρ₁)^0.5 = 0.8⋅(1.4)^0.5. Then, for each grid point in the region x < x_S = 0, the two following parameters are defined:

f(r) = V_m ⋅ exp{[1 – (r / r₀)²] / 2},   g(r) = 1 – f ² ⋅ [(γ–1)ρ₁] / (2γp₁) = 1 – f ²/7.

Finally, the flow parameters ahead of the shock wave are calculated as

u_x = u_x1 + f ⋅ (y–y_V) /  r₀,   u_y = u_y1 – f ⋅ (x–x_V) /  r₀,

ρ = ρ₁ ⋅ g^1/(γ–1) = g^2.5,    p = p₁ ⋅ g^γ/(γ–1) = g^3.5.

Dynamic viscosity coefficient is assumed constant and equal to μ = ρ₁u_x1(2r₀) / Re = 0.45⋅(1.4)^0.5⋅10^–4.

The flow is simulated until the time t = t₁ = 1.5 / u_x1 = 0.5⋅(1.4)^–0.5.

Note on start-up errors

The start-up errors (see, for example, [10, sec. 15.8.4], [9, sec. 3.3]) arise due to inconsistent (with numerical viscosity) smearing of the shock wave in the initial flow fields. Although such errors are local in nature, they can noticeably distort the complex structure of the flow in the considered problem. If the computational method used is sensitive to such errors, they can be easily eliminated by recovery the initial fields in the region x > x_S + 0.03 = 0.03 at the time t = t₁/6 (the vortex does not interact with the shock wave yet).